Given the thermodynamic and kinetic limitations which often constrain the extent of chemical reactions and post-reactional separation processes, and therefore constrain the yield and the degree of purity of the resulting products, integration of reaction and separation in a single unit has been under the scope of several bioengineering researchers in recent years. It is the aim of this work to compare the performance of a cascade of N reactor/separator sets, either in series or in parallel, with that of an integrated reaction/separation unit. In order to do so, a Michaelis-Menten reaction in dilute substrate solutions (i.e. a pseudo ®rst order reaction) was considered to take place in either con®guration and, under the same reaction and separation conditions, comparison of the performance and ef®ciency of these con®gurations was made in terms of fractional recovery of pure product, total time required to achieve such recovery and rate of recovery. It was concluded that: (i) the series combination of reactor/separator sets yields better results, both in terms of fractional amount of product recovered and time required to do so, than the parallel combination; and (ii) the integrated approach is much more timeand cost-effective than plain cascading, thus making it very attractive from an economic point of view. List of symbols A Interfacial area of the separator (m) C Molar concentration (mol mÿ3) i Generic i-th reactor/separator unit (i 1; 2; . . . ;N) kcat;f ®rst order intrinsic kinetic constant for the forward reaction (sÿ1) kcat;r ®rst order intrinsic kinetic constant for the reverse reaction (sÿ1) kmt Mass transfer coef®cient (m s ÿ1) Keq Equilibrium constant of the reaction (±) Km;S Dissociation constant of the enzyme/substrate complex (mol mÿ3) Km;P Dissociation constant of the enzyme/product complex (mol mÿ3) k1 Lumped ®rst order rate constant in the forward direction (sÿ1) kÿ1 Lumped ®rst order rate constant in the reverse direction (sÿ1) n Number of moles (mol) n̂ Number of moles at the outlet stream of the separator constituted by a mixture of R and P (mol) n Number of moles at the outlet stream of the separator constituted by pure P (mol) N Total number of reactor/separator sets S Substrate P Product r Reaction rate (mol mÿ3 sÿ1) t Time (s) v Molar volume (m molÿ1) V Volume of the reactor (m) vmax;f maximum rate of reaction in the forward reaction (mol mÿ3 sÿ1) vmax;r maximum rate of reaction in the reverse reaction (mol mÿ3 sÿ1) kcat;r ®rst order intrinsic kinetic constant for the reverse reaction (sÿ1) Greek letters a integration constant (±) v substrate conversion
v CP CS m algebraic stoichiometric coef®cient n extent of depletion by chemical reaction (±) f extent of depletion by physical separation (±) X dimensionless parameter (±) Subscripts eq equilibrium conditions E enzyme i generic reactor (i 1; 2; . . . ;N) L liquid phase P product par parallel combination rxn reaction S substrate ser series combination spn separation tot total A.L. Paiva, F.X. Malcata (&) Escola Superior de Biotecnologia, Universidade CatoÂlica Portuguesa, P-4200-072 Porto, Portugal The authors would like to acknowledge ®nancial support provided by JNICT through programs CIENCIA (grant BD/2081/ 92-IF, PhD fellowship) and PRAXIS XXI (grant BD/5568/95, PhD fellowship; and project 2/2.1/BIO/34/94 ± Extractive Biocatalysis, research grant). tot,N total number of reactor/separator sets 0 initial conditions Superscripts dimensionless variable 1 Introduction Transformation of substrate(s) into the corresponding product(s) is the aim of any reactional process. However, owing to thermodynamic and/or kinetic constraints, the yield and purity of the product(s) at the outlet of the reactor does not always attain the levels desired and, for this reason, biochemical reactors have been traditionally followed by separation units, designed in attempts to maximize such levels. Since the costs associated with post-reactional processing of biochemicals are usually high, alternative approaches, such as cascading sets of reactors/separators, have also been proposed and employed by many researchers when trying to improve the processing performance. Given that such analyses generally focus only on sets of reactors/separators used in sequential fashion without considering the hypothesis of their parallel use, one major goal of this work was, thus, to study whether cascading reactor/separator sets in parallel does (or does not) provide better results than in series. On the other hand, attempts to alleviate limiting factors, such as the high product inhibition and low volumetric productivity that are typical of biochemical processes, have led in the last two decades to comprehensive studies of integrated systems rather than sequential ones [1±7]. Integration can be seen as the limiting situation of cascading when the total number of sets tends to in®nite, and entails the possibility of continuously (and immediately) removing the product(s) formed during reaction and, consequently, improving effectiveness of separation since bulk concentrations are not allowed to build up. Such practice is, therefore, expected to reduce the aforementioned costs usually associated with post-reaction separation. Examples of integrated approaches which have recently been selected for biochemical processes encompass (but are not limited to) integrated liquid-liquid systems [8±10], integrated vapor-liquid systems [11±17], integrated supercritical ̄uid systems [18, 19], integrated solid-liquid systems [20±28] and integrated solid-gas systems [29±32]. A comprehensive review of applications of integration in biochemical processes has been provided elsewhere [33]. Laane et al. [1] and Tramper et al. [2] have claimed that reaction coupled with in situ separation brings about kinetic enhancements in the case of biochemical processes. However, such claim tends only to consider the point of view of the reaction rather than that of the overall process, constituted by both reaction and separation. Paiva and Malcata [4] have later demonstrated that integration of reaction and separation does not provide a true thermodynamic enhancement if Gibbs' free energy is used as quantitative measure because of its nature of state function (i.e. with changes that are independent of path). However, it was also demonstrated that, once physical separation is achieved on the molecular level right upon chemical reaction has taken place, integration decreases kinetic limitations via prevention of bulk mixing of product with (unreacted) substrate [6]. In spite of the complexity associated with modelling and prediction of behaviour in the case of integrated approaches, the actual decrease in the total manufacture cost of the product arising from lower reaction times (and thus lower capital investments in smaller reactors able to effect a given conversion of substrate) coupled with separation to a higher extent (and thus lower separation costs in a posteriori less intensive separation processes) may overcome this drawback. In order to maintain the analysis mathematically tractable throughout this study, the limiting ®rst order behaviour of a generic enzyme-catalyzed reaction was considered. The model system selected consists in a reaction that follows a 1:1 stoichiometry, which is in agreement with the current trend of bioprocess intensi®cation brought about by increasing substrate concentration to the highest degree possible (which also avoids use of solvents that add to downstream separation problems). The performance of a cascade of N similar reactor/separator sets (both in series and in parallel) was then compared with that of an integrated reaction/separation unit. Such comparison was made both in terms of the fractional amount of pure product recovered, its rate of recovery and total time (time of reaction plus time of separation) required to achieve the same extent of conversion and separation. 2 Mathematical development Let us consider a chemical transformation of substrate (S) into product (P) occuring via an enzyme-catalyzed reaction according to the following Michaelis-Menten reversible mechanism assumed to satisfy quasi-equilibrium conditions at all times: E S ! Km;S ES ! kcat;f kcat;r EP ! Km;P E P ;
1 where Km;S and Km;P are the dissociation constants of the enzyme/substrate complex and the enzyme/product complex, respectively, and kcat;f and kcat;r are ®rst order, intrinsic kinetic constants for the forward and reverse reaction, respectively. Under this postulated mechanism, the following rate expression can be derived [34]: r vmax;f CS Km;S ÿ vmax;r CP Km;P 1 CS Km;S CP Km;P ;
2 where r denotes the reaction rate, CS and CP represent the molar concentration of substrate and product, respectively, and vmax;f and vmax;r represent the maximum rates of reaction in the forward and reverse direction under saturation conditions of enzyme, and are given by: vmax;f kcat;f CE;tot; vmax;r kcat;rCE;tot ;
3 where CE;tot denotes the total concentration of catalytic sites; if the enzyme contains only one catalytic site per molecule, then CE;tot is equal to the total concentration of enzyme. If we now assume that both the concentrations of substrate and product are very small (and remain as such) when compared with the Michaelis-Menten parameters Km;S and Km;P, respectively, then their contribution to the denominator of Eq. (2) is negligible; this equation may thus be simpli®ed to: r k1CS ÿ kÿ1CP ;
4 which is typical of ®rst order reversible reactions, where kl and kÿl can be viewed as lumped ®rst order rate constants de®ned as vmax;f=Km;S and vmax;r=Km;P, respectively. One necessary condition for chemical equilibrium corresponds to r 0 which, in view of Eq. (4), implies that: k1CS;eq kÿ1CP;eq ;
5 where subscript eq denotes equilibrium conditions; upon algebraic rearrangement, Eq. (5) can take the form: k1 kÿ1 CP;eq CS;eq Keq ;
6 where Keq is the equilibrium constant of the reaction. Combination of Eqs. (4) and (6) then gives: r k1 CS ÿ CP Keq :
7 In view of the stoichiometry apparent in Eq. (1), one can also write: CS;0 CP;0 CS CP ;
8 where subscript 0 denotes initial conditions; if, as usual, no products are initially present (i.e. CP;0 0), then, at any time, Eq. (8) becomes: CP CS;0 ÿ CS :
9 Combination of Eqs. (7) and (9) yields: r k1 1 1 Keq CS ÿ k1CS;0 Keq :
10 Consider now a batch stirred tank reactor (which will eventually be a part of a combination of multiple identical reactors) where the aforementioned biochemical reaction is brought about. The mass balance to substrate S in said reactor takes the form:
ÿmSVirfCSg dnS dt 0 ;
11 where mS denotes the algebraic stoichiometric coef®cient of substrate S (equal to ÿ1 in our case), V denotes the (useful) volume of a reactor, subscript i refers to the i-th reactor, nS denotes the number of moles of S and t denotes the time of reaction. Using Eq. (10) in Eq. (11), one gets:

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